Calc 3 question

Question

Find Z_t and Z_p for Z = 4ysin(2x), x = 3u - P, y = (P^3)(u), u = t^2 + 1.

Answer 1

Ask yourself... what do I have here?

We have two independent variables essentially... p and t.

Then we have "functions" that are built on those variables:

"u" is a function of t... so let's write it that way... u(t)

"x" is a function of u and p... so let's write it that way... x(u,p)

"y" is also a function of u and p... so let's write it that way... y(u,p)

"Z" is a function of y and x...

When functions are embedded inside other functions, what does that typically mean in Calculus? We will need to use the "chain rule" for derivatives. Also, there are two independent variables so taking derivatives will be "partial derivatives".

Knowing the context will help us get to the solution.

Z_p means "Partial derivative of Z with respect to the variable p"

= (Z_y)*(y_p) + (Z_x)*(x_p)

Can you do those on your own?

Z_t = (Z_y)*(y_u)*(u_t) + (Z_x)*(x_u)*(u_t)

Try these on your own.